Abstract
Let N be a sufficiently large integer. In this paper, it is proved that, with at most \(O(N^{7/18+\varepsilon })\) exceptions, all even positive integers up to N can be represented in the form \(p_1^2+p_2^2+p_3^3+p_4^3+p_5^4+p_6^4\), where \(p_1,p_2,p_3,p_4,p_5,p_6\) are prime numbers, which constitutes an improvement over some previous work.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Brüdern, J.: Sums of squares and higher powers. J. Lond. Math. Soc. 35(2), 233–243 (1987)
Hua, L.K.: Additive Theory of Prime Numbers. American Mathematical Society, Providence (1965)
Liu, Y.H.: On a Waring–Goldbach problem involving squares, cubes and biquadrates. Bull. Korean Math. Soc. 55(6), 1659–1666 (2018)
Lü, X.D.: Waring–Goldbach problem: two squares, two cubes and two biquadrates. Chin. Ann. Math. Ser. A 36(2), 161–174 (2015)
Lü, X.D.: On unequal powers of primes and powers of 2. Ramanujan J. 50(1), 111–121 (2019)
Pan, C.D., Pan, C.B.: Goldbach Conjecture. Science Press, Beijing (1981)
Ren, X.M.: On exponential sums over primes and application in Waring–Goldbach problem. Sci. China Ser. A 48(6), 785–797 (2005)
Vaughan, R.C.: On the representation of numbers as sums of squares, cubes and fourth powers and on the representation of numbers as sums of powers of primes. Philosophy Doctor Degree Thesis, University College London (University of London) (1970)
Vaughan, R.C.: The Hardy–Littlewood Method, 2nd edn. Cambridge University Press, Cambridge (1997)
Wooley, T.D.: Slim exceptional sets and the asymptotic formula in Waring’s problem. Math. Proc. Camb. Philos. Soc. 134(2), 193–206 (2003)
Zhang, M., Li, J.: Exceptional set for sums of unlike powers of primes. Taiwan. J. Math. 22(4), 779–811 (2018)
Zhao, L.: On the Waring–Goldbach problem for fourth and sixth powers. Proc. Lond. Math. Soc. 108(6), 1593–1622 (2014)
Acknowledgements
The authors would like to express the most sincere gratitude to the referee for his/her patience in refereeing this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work is supported by the Fundamental Research Funds for the Central Universities (Grant No. 2019QS02), and National Natural Science Foundation of China (Grant Nos. 11901566, 11971476).
Rights and permissions
About this article
Cite this article
Zhang, M., Li, J. Exceptional set for sums of unlike powers of primes (II) . Ramanujan J 55, 131–140 (2021). https://doi.org/10.1007/s11139-020-00252-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-020-00252-3